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Problem 4 A discrete random variable X follows the geometric distribution with parameter p, written X ~Geom(p), if its distribution function is fx(x) = p(1-p)"-1, xe(1, 2, 3, . . .} The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that Ix(z) is indeed a probability inass function, i.e., the sum over all possible values of z is one b) Let |c <1, and consider the sum By multiplying S by c, we find By combining these two expressions, prove that S 1/(1 - c)2 c) Prove that the expected value of X ~Geom(p) is 1/p. Hint: the result of the previous problem might be useful d) Let ri,..., Xn be n observations of a geometric random variable with parameter p. Prove that the maximum likelihood estimator for the parameter p is Σ|-1 xi

Answer 1

Here, we check the given function is probability mass function and further find the mean and maximum likelihood estimator of p of geometric distribution.
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